Interpretation of "If A, then B" as "A coupled with B": Context and Applicability

Question

It is known that the meaning of a conditional statement in fuzzy logic can vary depending on the interpretation and context. In certain fuzzy logic books, I have come across the interpretation that "if A, then B" can be understood as "A coupled with B" when it means $\mu(x,y) = T_{norm} (\mu_{A}(x), \mu_{B}(y))$, instead of $\neg A \lor B$.

I would like to know:

• in which context this interpretation is applicable?
• why we have to use $A \Rightarrow B$ notation and symbols when it means $A \land B$?
• whether it has any relevance in classical logic?
• Furthermore, I am curious to explore whether this interpretation can be rational and relevant in everyday human speech?

Update

Mamdani in An experiment in linguistic synthesis with a fuzzy logic controller has said:

The control rules were implemented by using fuzzy conditional statements, for example "If PE is $NB$ then HC is $PB$". The implied relation between the two fuzzy variables PE and HC is expressed in terms of the Cartesian product of the two subsets $NB$ and $PB$. The Cartesian product of two sets $A$ and $B$ is denoted $A \times B$ and is defined by $$ A \times B = \sum\limits_{i} \sum\limits_{j} \min \{ \mu_{A}(u_i), \mu_{B}(v_j) \} \\ = \sum\limits_{ij} \min u_i v_j $$ where $u$ and $v$ are generic elements of the universes of discourse of $A$ and $B$ respectively.

As you can see, Mamdani used $\min (\mu_{A}(x), \mu_{B}(y))$ as interpretation for $A \rightarrow B$, which is

• equal to $A \land B$
• a special case for $T_{norm} (\mu_{A}(x), \mu_{B}(y))$
• not equal to $\neg A \lor B$

In some books, for example "Neuro-Fuzzy and Soft Computing: A computational Approach to Learning and Machine Intelligence" the $T_{norm} (\mu_{A}(x), \mu_{B}(y))$ interpretation is called as "A coupled with B".
Although, here the word "coupled" is not so important for me. I wanted to know when that interpretation is rational?

Answer 1

OP is asking for clarification about the contents of some text books , hence it is a valid query.
The hitch here is that these are not mainstream text books with mainstream thinking : hence there are not many comparable text books where we can get clarifications.
With that , these are my equally non-mainstream thinking on the matter.

OP Doubts : I would like to know:

• in which context this interpretation is applicable?
  -- It is generally not applicable. If at all it can be used , it will be in non-critical cases like recommending books & videos [[ I will explain more in Example Section ]]
• why we have to use $A \Rightarrow B$ notation and symbols when it means $A \land B$?
  -- The Author is trying to convert classical logic ( where implication is a core concept ) to fuzzy logic ( where the authors are claiming that we do not require the implication at all )
• whether it has any relevance in classical logic?
  -- It is eminently unsuitable for the rigorous models of classical logic
• Furthermore, I am curious to explore whether this interpretation can be rational and relevant in everyday human speech?
  -- It is not relevant in general. We can try to force-fit it to human communication , which seems to be what the authors are indirectly claiming [[ I will explain more in Example Section ]]

Example :

Consider some user looking at mathematics text books [ on amazon.com ] or viewing music videos [ on youtube.com ] : It is non-critical to suggest other text books or other videos.
We can think in two ways like this :
(1) When user sees or buys "theoretical statistics" , then user might want to see or buy "applied statistics" , which is like Implication here.
(2) Here , user saw "theoretical statistics" , lots of earlier users both "theoretical statistics" AND "applied statistics" , hence user might want to see "applied statistics" too , which is like AND.
We can convert the Implication to AND , without logical justification though there is no harm & it is non-critical.

In a way , "theoretical statistics" is coupled with "applied statistics" , where both will occur when one occurs.

We do not know what was the reason to buy both :

• maybe earlier users wanted to buy both [[ A AND B ]] to attended some course
• maybe one book (theory) was prerequisite for the users to read the other (applied) to know the usage [[ A IMPLIES B ]]
• maybe users wanted to one (applied) but added the other (theory) to ensure rigor in the usage [[ B IMPLIES A ]]

This type of fuzzy thinking is okay in the non-critical contexts like music recommendations & hotel reviews & elevator scheduling.
This is not okay in the critical contexts like medical diagnosis & satellite controllers & criminal proceedings where scientific accuracy & mathematical precision & logical rigor are necessary.

In Statistics , this vague concept is known as "correlation is not causation" : We can replace $A \land B$ with neither $A \implies B$ nor $B \implies A$ in classical logic & in everyday cases too.

In general , fuzzy logic & fuzzy thinking are not very widely applicable.